Year: 2026
Type: preprint
Abstract: We study the finite basis problem for flat semirings of the forms \(S_c(W)\), \(M_c(W)\), and \(M(W)\), where \(W\) is a nonempty set of words in a free commutative semigroup, a free commutative monoid, and a free monoid, respectively. We completely classify such flat semirings with respect to the finite basis property, allowing \(W\) to be infinite. We prove that \(S_c(W)\) is finitely based if and only if every word in \(W\) is either a cube of a letter or has length at most two, whereas \(M_c(W)\) and \(M(W)\) are finitely based if and only if \(W\) consists solely of the empty word. As applications, we recover the nonfinite basability of \(\flat(\mathbb{Z})\) and the max-plus semiring \((\mathbb{Z},\max,+)\).
Source: arXiv (Cornell University)
Authors: Gao, Zidong, Ren, Miaomiao, Zhou, Yilin
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: G3090029520, G7299353239
OpenAlex ID: W7215089424